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Submitted on 13 Jan 2021 (v2), last revised 15 Apr 2021 (v3)

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Thermodynamic Effects on Grade Transition ofPolyethylene Polymerization in Fluidized Bed Reactors

Sabrine Kardous, Timothy Mckenna, Nida Sheibat-Othman

To cite this version:Sabrine Kardous, Timothy Mckenna, Nida Sheibat-Othman. Thermodynamic Effects on Grade Transi-tion of Polyethylene Polymerization in Fluidized Bed Reactors. Macromolecular Reaction Engineering,Wiley-VCH Verlag, 2020, pp.2000013. �10.1002/mren.202000013�. �hal-02749293v2�

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Thermodynamic effects on grade transition of polyethylene polymerization

in fluidized bed reactors

Sabrine Kardous1, Timothy F.L. McKenna2, Nida Sheibat-Othman1*

1 Université of Lyon, Université Claude Bernard Lyon 1, CNRS, UMR 5007, LAGEPP, F-

69100, Villeurbanne, France

2 Université of Lyon, Université Claude Bernard Lyon 1, CPE Lyon, CNRS, UMR 5265,

C2P2 - LCPP group, Villeurbanne, France.

*E-mail : nida.othman@univ-lyon1.fr

Graphical abstract

Abstract

An off-line dynamic optimization procedure is employed to optimize the transition between

different grades of linear low density polyethylene in a fluidized-bed reactor. This type of

reactor is frequently operated under condensed mode, which consists of injecting induced

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condensing agents (ICA) to absorb part of the reaction heat. However, the presence of ICA

affects the solubility of monomers in the polymer, so it is important to account for this effect in

a grade transition optimization strategy. A kinetic model is combined with a thermodynamic

model based on the Sanchez-Lacombe equation of state to describe the grade transitions.

Simplified correlations are then suggested to predict the impact of ICA on ethylene and

comonomer solubility in a quaternary system. The results highlight the importance of the

thermodynamic model during grade transition.

Keywords

Grade transition, fluidized bed reactor, condensed mode cooling, thermodynamics,

polyethylene.

1. Introduction

Polyethylene (PE) is the most widely produced polymer in the world. Among the different types

of PE, linear low density polyethylene (LLDPE) occupies an important position with around

30% of the global PE production in 2018.[1] Most processes used to make LLDPE are gas-phase

processes which provide several advantages over slurry processes. Indeed, difficulties related

to mass transfer limitations and the dissolution of the amorphous polymer in a diluent, as well

as fouling in slurry processes are avoided in gas-phase processes. Gas-phase processes are

adequate for multipurpose production and permit the production of a wide range of PE grades.

Among gas-phase processes, fluidized-bed reactors (FBRs) are far and away the most widely

used reactors for the production of LLDPE because they are the only reactors that can remove

enough heat in the gas phase and thus allow the production of large amounts of polymer.[2] In

order to further enhance heat transfer and increase productivity, condensed mode cooling is

frequently employed, where induced condensed agents (ICAs), which are typically alkanes such

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3

propane or isomers of butane, pentane or hexane, are injected in either liquid or vapor form.[3,4]

The heat of vaporization and/or increase in the heat capacity of the vapor phase in the reactor

absorb a significant amount of the reaction heat and improve the control of the reactor

temperature. However, it has also been observed that when the polymer particles are swollen

by an alkane or an alkene, the reaction rate can change significantly due to the so-called

cosolubility effect. Indeed, the presence of a hydrocarbon heavier than ethylene enhances the

solubility of the latter in the amorphous phase of the polymer, thereby contributing to a higher

rate of polymerization, while the lighter hydrocarbons play the role of anti-solvent for the

heavier ones.[5,6] Therefore, the presence of ICA increases the ethylene concentration in the

particles, leading to a higher reaction rate, while ethylene is expected to act as an anti-solvent

for the ICAs (and eventually for comonomers like 1-butene or 1-hexene). This is expected to

impact the properties of the final product such as its molecular weight and density. Therefore,

it is important to account for these effects in the process model, especially in model-based

optimization or control strategies.

It is quite common to produce several grades in the same polymerization plant in order to obtain

a PE with different density, molecular weight and polydispersity index required in the various

applications of PE.[7] Frequent transitions between these grades are usually needed to suit the

market demand and reduce the storage cost. Due to the long residence time in FBRs, compared

to tubular or loop reactors for instance, the flow rates should be optimized wisely to ensure

attaining the new set-point in a short time, thus reducing the amount of transition product. When

condensed mode is employed, the transitions might be more complex since the

sorption/desorption dynamics of the different species can change the behavior of the system.

The employment of an adapted control of the transition is thus essential in order to optimize the

economic yield while ensuring the security of the operations.

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4

Debling et al.[7] studied the effect of different parameters on grade transition of solution, slurry,

bulk and gas-phase polyolefin reactions in commonly used reactors, including horizontal or

vertical stirred beds, loop and FBRs. They indicated the residence time distribution of the

different components to be a determining factor in the speed of grade transition and summarized

the procedures employed to speed the transition in FBRs, such as de-inventorying the reactor

content or venting/overshooting the gases at the beginning of the transition. However, they did

not investigate the effect of ICA on the residence time of the reactor. Rahimpour et al.[8] also

indicated that partial venting of the reactor, composing a new gas phase and reducing the bed

level reduce the quantity of transition product in PE FBRs. They highlighted that such so-called

semi-continuous strategy was necessary in some situations in order to keep the reactor

temperature between the gas dew point and the polymer melting point (to avoid agglomeration

of the particles), which could not be achieved with the continuous strategy (i.e. by controlling

only the flow rates). Note that the flow rates employed during the transition were those used

for the final grade, which were calculated by solving the model equations under steady state

conditions, and identifying the boundary conditions to be implemented for each new grade.

However, numerous works indicated that the flow rates of the final grade do not necessarily

ensure the best transition, and suggested the employment of dynamic optimization or control

algorithms to ensure better transitions. [9,10]

For the particular problem of grade transition in FBRs, most works are based on offline

optimization due to the long calculation time and the complexity of problem formulation.

McAuley et al.[11,12] were the first to investigate dynamic optimization of grade transition of PE

in gas-phase FBR. They provided a kinetic model for copolymerization, correlations for the

final properties based on patent data (i.e. melt index and polymer density), and modelled the

FBR as a continuous stirred tank reactor (CSTR) due to its high recycle ratio and low single

pass conversion. The suggested control variables were the flow rates of hydrogen and

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comonomer (1-butene or 1-hexene), as they directly affect the polymer molecular weight and

density. Afterwards, optimization strategies were proposed for different types of processes, for

instance for slurry high density polyethylene (HDPE) processes composed of two loop

reactors[13] or two CSTRs.[14]

Furthermore, different improvements in the optimization approaches were suggested. For

instance, Chatzidoukas et al.[15] proposed a mixed integer dynamic optimization approach to

realize a closed-loop control in a fluidized-bed reactor. Nystrom et al.[16] employed a

comparable approach based on dynamic optimization combined to a mixed-integer linear

problem related to the sequencing, and solved these decoupled problems by iteration. Bonvin

et al.[17] proposed to employ a measurement-based approach by tracking the necessary

conditions of optimality, for instance based on run-to-run basis, in order to correct for modelling

mismatch in a homopolymerization process.

Regarding closed-loop control, it was usually considered using algorithms based on an

optimization criterion, such as model predictive control (MPC). The closed-loop character of

MPC makes it more robust to modelling errors than open-loop dynamic optimization. But, in

order to allow its online implementation in FBRs, part of the optimization is usually solved

offline. For instance, Wang et al.[18] combined an offline optimizer and a nonlinear MPC, where

the optimal feed rates were calculated offline and the MPC allowed minimizing the modeling

error and updating the feed rates. A shrinking horizon nonlinear model predictive control with

expanding horizon least-squares estimation was also implemented to control the grade

transition in FBRs. [19]

Among all the cited works, in terms of methodology, dynamic optimization-based policies were

the most widely used for gas-phase processes, and they will therefore be employed in this work.

Indeed, the optimization criterion is more flexible and can be tuned to optimize instantaneous

or cumulative properties during transition, or the transition time. In addition, the previous

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6

literature analysis highlights that the parameters that most affect the grade transition are the

residence time (distribution) and the hydrogen and comonomer flow rates. However, the

thermodynamic interactions that are due to the use of a condensing agent and/or comonomer

were not considered. This work is focused particularly on condensed mode, and will explore

the implications of using a more representative thermodynamic model than most works that

takes into account the interactions between the different species. It is clear that simply using

additive solubilities will lead to erroneous conclusions about reaction rates and product

properties. It is thus important to understand whether or not this is an important part in

optimizing grade transitions.

In this work, the grade transition of PE copolymers in a gas-phase FBR operating under

condensed mode is considered. First the effect of ICA (n-hexane or iso-butane) on the

absorption of monomer (ethylene) and comonomer (1-butene or 1-hexene) is investigated using

a model based on the Sanchez-Lacombe equation of state (SL EOS) and experimental data from

literature.[20,21] Since no sufficient experimental data are available in the open literature for

quaternary systems (i.e. a copolymerization in presence of an ICA), simplifying correlations

are proposed in order to allow for fast prediction of the co-solubility effects in a quaternary

system.[21] The thermodynamic correlations are then used to calculate equilibrium solubilities

for two copolymerization systems of ethylene with -olefins. The thermodynamic predictions

are combined with kinetic copolymerization models to model the dynamic behavior of the FBR,

which is assumed to behave like a single-phase CSTR. The model is valid in the super-dry

upper compartment of the FBR, containing only gas and polymer. This dynamic model is finally

used within a dynamic optimization strategy to optimize the transitions between different grades

of LLDPE.

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7

2. Gas-phase catalytic ethylene co-polymerization model

2.1 Thermodynamic model

The SL EOS has been used frequently to predict the thermodynamic behavior in binary and

ternary systems (polymer plus one or two penetrants respectively). [22,23] However, as discussed

by McKenna[3], solubility data for systems of two penetrants is very scarce, and realistic data

for multiple penetrants is totally absent from the open literature. Therefore, in the absence of a

reliable data set, we will propose a simplified model accounting for 3 penetrants (i.e. quaternary

system) in an LLDPE plant. Two systems are considered, at 90°𝐶, which are frequently

employed in industry:

1. Copolymerization of ethylene and 1-hexene in presence of n-hexane as ICA

2. Copolymerization of ethylene and 1-butene in presence of iso-butane as ICA

In the Sanchez-Lacombe EOS, the concentration of the different components in the polymer is

predicted based on the binary interaction parameters 𝑘 between each pair of components

(the penetrating species and the polymer). The thermodynamic model is based on the following

assumptions: i) The gases dissolve only in the amorphous phase[24]; ii) The interaction between

molecules of olefins and/or ICA is ideal.[25] Thus, in a quaternary system of three penetrating

gaseous species, ethylene (1), ICA (2) and comonomer (3), in the polymer (4), 𝑘 , 𝑘 and 𝑘

are equal to zero. Finally, only the global degree of crystallinity of the polymer is accounted for

in the thermodynamic model. Strictly speaking, also tie molecules linking the crystalline

lamellae can influence the solubility of different species in the amorphous phase.[26]

Due to the lack of solubility data, the following, more critical assumptions are considered:

A.1 Additive effect: The quaternary system is approximated by a ternary system: ethylene/

(ICA+comonomer)/ LLDPE, as suggested by Alves et al.[21] Thus, a “pseudo” component,

representing the mixture of ICA plus comonomer, is defined for which the thermodynamic

8

8

parameters are identified. In this assumption, no interaction between ICA and comonomer

is considered, which means that these species behave independently from each other as if

they were present in a ternary system (PE, ethylene and either ICA or comonomer). This is

not unreasonable if the comonomer and ICA are similar in structure. This assumption is thus

applicable for the two systems studied in this work, i.e the comonomer 1-hexene and n-

hexane as ICA, as well as the comonomer 1-butene and iso-butane as ICA. However, this

assumption does not mean that the ICA and the comonomer have the same solubility or co-

solubility effect, as discussed in the following two sections.

A2. Polynomial approximations: In order to reduce the computation time, the results from the

SL EOS or the experimental results, are approximated by polynomials of degree 1 or 2, as

suggested by Alves et al.[21] Different pathways were considered for the two systems

investigated in the present work, as described in the following sections.

2.1.1 Polyethylene, in presence of ethylene, 1-hexene and the ICA n-hexane

For the first system, the comonomer 1-hexene and the ICA n-hexane, both the comonomer and

the ICA were found to have comparable solubilities in a binary system, especially at low

pressure, as shown by Figure 1 (Yao et al.[27] and Jin et al.[28]). Therefore, Alizadeh et al.[23]

assumed it safe to consider that they have similar solubility in LLDPE in a ternary or quaternary

system. A similar observation was found for other 3 and 6 carbon pairs, such as the isomers

propene and propane, where the difference in their solubility constant of Henry’s law was 10%

at 25°C, as reported by Michaels et al.[24]. This can be explained by the fact that 1-hexene and

n-hexane have similar shapes and same number of carbons, so almost the same size, therefore,

they have similar tendency to condense (i.e., same volatility). Besides, they have similar nature

of interaction with LLDPE segments (i.e., same nature of van der Waals forces).[23] Therefore,

in a quaternary system, the ratio of solubility of 1-hexene in LLDPE to n-hexane is assumed

𝑟

1. Note that for the second system, the comonomer 1-butene and the ICA iso-

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butane, their binary solubilities in LLDPE are different, and was accounted for as explained in

section 2.1.2.

Figure 1. Solubility of n-hexane and 1-hexene in LLDPE (binary systems) at 70°C (data from Yao et al.[27] and Jin et al. [28])

For this system, copolymerization of ethylene with 1-hexene in presence of the ICA n-hexane,

solubility data are available only for the ternary system ethylene/n-hexane/PE at 10 bar

ethylene[27] (Figure 2). To use these data in a quaternary system (Assumption A1: Additive

effect), the ICA n-hexane is assumed to thermodynamically behave like the comonomer 1-

hexene (as explained in the previous section).[23]

The available ternary data of solubility (Figure 2) is then used to obtain linear or polynomial

equations (Assumption A2: Polynomial approximations). It can be noticed that the solubility

of ethylene varies linearly over a small range with the pressure of ICA/comonomer, therefore a

polynomial of degree 1 could fit the experimental data, while the solubility of comonomer

varies nonlinearly with its pressure, therefore a polynomial of degree 2 was necessary, as

follows:

0 0.2 0.4 0.6 0.8

n-hexane or 1-hexene Partial Pressure(bar)

0

0.02

0.04

0.06

0.08

0.1

0.12

So

lub

ility

(g/g

am L

LD

PE

)

Yao et. al.,70°C (n-hexane/LLDPE)Jin et. al.,70°C (1-hexene/LLDPE-hexene)Jin et. al.,70°C (1-hexene/LLDPE-butene)Jin et. al.,70°C (1-hexene/LLDPE-octene)

10

10

M M , 𝐴 𝑃 𝑃 𝐵 (1)

M M , 𝑟𝑃

𝑃 𝑃𝐶 𝑃 𝑃 𝐷 𝑃 𝑃 (2)

Where M , and 𝑃 are respectively the concentration in amorphous polymer (mol m-3) and

pressure (bar) of component 𝑖 (1 for ethylene and 2 for the comonomer) and r represents the

ratio of solubility of comonomer to ICA, which is equal to one in this system. The values of the

coefficients (A, B, C, D) in equations 1 and 2 are given in Table 1. The concentration of

monomer in the bed is M M , 1 𝑥 , where 𝑥 is the polymer crystallinity. It is

important to mention that these coefficients are valid for the specific conditions (temperature

and pressure) for which they were developed (i.e. ethylene pressure of 10 bars and pseudo-

component ‘ICA+comonomer’ pressure of 0 to 1 bar). It is also important to note that the

concentration of ethylene only varies slightly as a function of the partial pressure of “n-hexane

+1-hexene”, however the concentration of n-hexane (and so of 1-hexene) is very sensitive to

the partial pressure of the pseudo-component ‘ICA+comonomer’, which is necessary to account

for in the model (Figure 2). Note that when increasing temperature, the slope of the

concentration of n-hexane with its partial pressure decreases (from C 624 to 506 mol m-3 Pa

-2), while the slope of the concentration of ethylene increases (from A 34 to 279 mol m-3 Pa -

1). Increasing the temperature thus increases slightly the impact of ICA on ethylene absorption,

but it remains very low. Besides, the figure shows that the concentration of n-hexane in

amorphous PE increases slightly in a binary system compared to a ternary system. This was

expected given the anti-solvent effect of ethylene on ICA in a ternary system.[29,30] Note that

the concentration plots presented in this paper were calculated by converting solubility data (in

g g-1 amorphous polymer) found in the open literature into mol m-3 using the amorphous

polymer densities (i.e., the values of the swollen polymer density with different ICAs) estimated

by Sanchez-Lacombe EoS for each case.

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Figure 2. Ternary solubility data ethylene/n-hexane/LLDPE at 90°C and 70°C at 10 bar of ethylene (experimental data are taken for ternary systems from Yao et al.[31] and for binary systems from Yao et al.[27]): a) Concentration of ethylene in amorphous LLDPE as a function of the partial pressure

of n-hexane, b) Concentration of n-hexane as a function of its partial pressure.

Table 1. Coefficients of the correlations allowing to estimate the ethylene and 1-hexene concentrations in amorphous LLDPE at 90 °C and 70 °C (valid at ethylene pressure of 10 bar and

pseudo-component pressure on the range 0-1 bar)

Value at

90°C

Value at

70°C

units

A 33.8 25.9 mol m-3 bar -1

B 251 278.8 mol m-3

C 505.8 623.9 mol m-3 bar -2

D 169.2 1206.3 mol m-3 bar -1

r 1 1 -

Equations 1 and 2, with their identified parameters in Table 1, allow for the estimation of the

concentration of ethylene and 1-hexene in the quaternary system ethylene/1-hexene/n-

0 0.2 0.4 0.6 0.8 1

ICA Partial Pressure (bar)

250

260

270

280

290

300

Cet

,po

lym

er(m

ol m

-3)

Ternary-T=90°CBinary-T=90°CTernary-T=70°CBinary-T=70°CLinear approximation

0 0.2 0.4 0.6 0.8 1

ICA Partial Pressure (bar)

0

200

400

600

800

1000

Cn

-hex

ane,

po

lym

er(m

ol m

-3)

Ternary-T=90°CTernary-T=70°CBinary-T=90°CBinary-T=70°CPolynomial approximation

ba

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12

hexane/LLDPE at equilibrium. This information is all what is required for the rest of the paper

regarding this system, and there is no need for the SL EoS here.

2.1.2 Polyethylene, in presence of ethylene, 1-butene and iso-butane as ICA

For the copolymerization of ethylene with 1-butene in presence of iso-butane as ICA, ternary

data is not available for either ethylene/iso-butane/LLDPE or for ethylene/1-butene/LLDPE.

Moreover, the available binary sorption data (Figure 3) [32,33] indicates that 1-butene and iso-

butane have different solubilities in the polymer and cannot be assumed to be similar, as could

be done for 1-butene/n-butane and 1-hexene/n-hexane. Indeed, the solubility of 1-butene in

HDPE is higher than that of iso-butane by almost a factor of 𝑟

1.78, at nearly

the same temperature.

Figure 3. Binary solubility data in HDPE of a) comonomer 1-butene at 𝟔𝟖. 𝟗°𝑪, and b) ICA iso-butane at 65.6°C.

Due to this lack of data, the following assumptions are made only for this system:

A3. The ratio of sorption of 1-butene to iso-butane, 𝑟 (Figure 3), identified in a binary system,

was assumed to remain unchanged in a ternary system, and to remain unchanged within a

0 5 10

1-Butene Pressure (bar)

0

500

1000

1500

2000

2500

Co

nce

ntr

atio

n1-

Bu

ten

e,p

oly

mer

(mo

l/m3) Moore and Wanke,2001

SL EoS kij=0.017

0 5 10

Iso-Butane Pressure (bar)

0

500

1000

1500

2000

2500

Co

nce

ntr

atio

nIs

o-B

uta

ne,

po

lym

er(m

ol/

m3)

Parrish et al.,1981SL EoS kij=0.017

a b

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13

temperature range of 65-90 °C. This means that they are assumed to have the same co-

solubility effect on ethylene.

A4. The 𝑘 parameters are assumed to vary linearly with temperature. So, as solubility data is

not available, an extrapolation of 𝑘 is done to predict them at 90°𝐶.

A5. The solubility of iso-butane in HDPE and LLDPE is assumed to be the same, as only

solubility data in HDPE is available (in a binary system[32]).

Again, Assumption A1 is applied to this system: The comonomer and ICA were assumed to

have an additive effect in a quaternary system. Alves et al.[21] validated this assumption for

propane and iso-butane based on reaction data from patents. This means that propane and iso-

butane do not affect the solubility of each other (i.e. there is no co-solubility effect), which is

reasonable in view of their similarities. A similar assumption can be done regarding 1-butene

and iso-butane in our system. Based on this assumption, Alves et al.[21] estimated the binary

interaction parameters 𝑘 for the ternary system ethylene/iso-butane/LLDPE by fitting to

experimental solubility data of the binary systems ethylene/LLDPE and iso-butane/HDPE at

70°C (Table 2).[32] Thus, the correlation between the ICA and the comonomer becomes

(combining Assumptions A1, A2 and A3):

M , 𝑟𝑃

𝑃 𝑃𝐸 𝑃 𝑃 (3)

In order to identify the parameters of equations 1 and 3 for the quaternary system, SL EOS is

first used in the ternary system ethylene/iso-butane/LLDPE (with 𝑘 identified using binary

solubility data [21]) to identify the solubility of ethylene and iso-butane at different pressures of

iso-butane. Then, the 𝑘 parameters were extrapolated over temperature to have values at 90°C

(Assumption A4) (Figure 4, Table 3). The concentration of 1-butene is then calculated using

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𝑆 𝑟 𝑆 . From the obtained data points, polynomials of order 1 were

identified for both ternary systems.

As in the first system, it can be noted that the concentration of ethylene only varies slightly as

a function of the partial pressure of the pseudo-component “isobutane+1-butene”, while the

concentration of comonomer 1-butene is very sensitive to its partial pressure (Figure 4).

Table 2. Binary interaction parameters of the ternary system ethylene/iso-butane/LLDPE (based on binary thermodynamic data)

Temperature °𝐶 Ethylene/iso-

butane 𝑘

Ethylene/LLDPE

𝑘 23

Iso-butane/LLDPE

𝑘

70°C 0 -0.014 0.025 (74°C) [21]

80°C 0 -0.022 0.022 (82°C) [21]

90°C 0 -0.032 -3.75 10-4 T(K)+0.1551

Figure 4. Concentrations in amorphous LLDPE obtained using SL EoS, Pseudo-quaternary system ethylene/(ICA+comonomer)/HDPE, at 𝟗𝟎°𝑪 (after extrapolation of 𝒌𝒊𝒋) and 7 bar of ethylene: a)

ethylene and b) comonomer 1-butene.

0 5 10

ICA+comonomer Partial Pressure(bar)

0

50

100

150

Cet

,po

lym

er(m

ol m

-3)

SL EoS data at 90°CLinear approximation

0 5 10

ICA+comonomer Partial Pressure(bar)

0

300

600

900

C1-

bu

ten

e,p

oly

mer

(mo

l m-3

)

SL EoS data at 90°CLinear approximation

a b

15

15

Table 3. Coefficients of the correlations allowing to estimate the ethylene and 1-butene concentrations in amorphous LLDPE at 𝟗𝟎°𝑪 (valid at ethylene pressure of 7 bar and pseudo-

component pressure on the range 5-10 bar)

Parameter at 𝟗𝟎°𝑪 Units

A 0.992 mol m-3 Pa -1

B 134.73 mol m-3

E 90.209 mol m-3 Pa -1

r 1.78 -

2.2 Kinetic model

A classical reaction scheme (Table 4) of copolymerization of ethylene in presence of a catalyst

having one type of active sites was considered (see for instance de Carvalho et al.[34], McAuley

et al.[35], Chatzidoukas et al.[15]). The only difference between the two copolymerization

systems, with the comonomer 1-hexene or with 1-butene, is the values of the rate constants.

The following notations are used in Table 4: S : potential catalyst active sites, P : activated

vacant catalyst sites, P∗: total active sites (vacant P and occupied by a polymer chain P , ), P , :

living copolymer chains of length n ending with monomer i, D : dead copolymer chains of

length n and C : deactivated catalyst sites.

The reaction rates resulting from the proposed reaction scheme are given in Table 5. All

concentrations are in (mol m-3) and the following notations are used: λ moment 𝑘 of living

chains, 𝜇 moment 𝑘 of dead chains and 𝑀 , is the cumulative polymer average molecular

weight.

Table 4. Kinetic scheme of the copolymerization of ethylene with a catalyst of one site (without co-catalyst)

Designation Reaction

Spontaneous activation S → P

Chain initiation P M P ,

Propagation P , M ⎯ P ,

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Spontaneous deactivation P , ⎯ C +D , P ⎯ C

Spontaneous chain transfer P , ⎯ P +D

Chain transfer to hydrogen H P , H P +D

Chain transfer to monomer M P , M ⎯⎯ P , +D

Table 5. Reaction rates of the different species (mol m-3 s-1)

𝑅 𝑘 S

𝑅 𝑘 S 𝑘 P 𝑘 M 𝑘 M P 𝑘 𝑘 H 𝜆

𝑅 P 𝑘 M 𝑘 M 𝜆 𝑘 𝑘 𝑘 H

𝑅 P 𝑘 M 𝑘 M 𝜆 𝑘 𝜙 𝑘 𝜙 𝑘 𝜙

𝑘 𝜙 M 𝑘 𝜙 𝑘 𝜙 𝑘 𝜙 𝑘 𝜙 M 𝜆 𝑘 𝑘

𝑘 H 𝑘 𝜙 𝑘 𝜙 M 𝑘 𝜙 𝑘 𝜙 M

𝑅 P 𝑘 M 𝑘 M λ 𝑘 𝜙 𝑘 𝜙 M 𝑘 𝜙

𝑘 𝜙 M λ 2λ 𝑘 𝜙 𝑘 𝜙 M 𝑘 𝜙 𝑘 𝜙 M

𝜆 𝑘 𝑘 𝑘 H 𝑘 𝜙 k 𝜙 M 𝑘 𝜙 𝑘 𝜙 M

𝑅 λ 𝑘 𝑘 𝑘 H 𝑘 𝜙 𝑘 𝜙 M 𝑘 𝜙

𝑘 𝜙 M

𝑅 λ 𝑘 𝑘 𝑘 H 𝑘 𝜙 𝑘 𝜙 M 𝑘 𝜙

𝑘 𝜙 M

𝑅 λ 𝑘 𝑘 𝑘 H 𝑘 𝜙 𝑘 𝜙 M 𝑘 𝜙

𝑘 𝜙 M

𝑅 𝑘 H 𝜆

𝑅 𝑘 P 𝑘 𝜙 𝑘 𝜙 𝑘 𝜙 𝑘 𝜙 𝜆 M

𝑅 𝑘 P 𝑘 𝜙 𝑘 𝜙 𝑘 𝜙 𝑘 𝜙 𝜆 M

With 𝜙 , 𝜙 1 𝜙 ,

H , (mol m-3 am. polymer)=𝑆 , , 𝑆 (g H2 g-1 am. polymer)=10-9𝑃 [36]

17

17

𝜆 𝜆 , 𝜆 , ∑ 𝑛 P , ∑ 𝑛 P , ,

𝜇 ∑ 𝑛 D

𝑀 , 𝑀 , , 𝑀 , 𝑀 , , with 𝑀 , ∑ 𝜑 𝑀 , 𝜑∑

The values of the different kinetic rate constants are given in Tables 6 and 7. 𝐸 refers to the

activation energy and k0 to the pre-exponential factor. For the case of ethylene-co-1-butene, the

parameters were taken from Chatzidoukas et al.[15] or Ghasem et al.[37]. For the system ethylene-

co-1-hexene in gas-phase, fewer parameters are available. Chakravarti et al.[38] gave some

kinetic parameters for this system using a metallocene catalyst. In order to keep both systems

comparable in terms of catalyst activity, only the reactivity ratios were taken from Chakravarti

et al.[38], and the other parameters and ratios were kept as for the first system. The identification

of a specific kinetic model for a defined system is out of the scope of this work as our primary

focus is to explore the impact of the co-solubility effect on grade change optimization.

Table 6. Pre-exponential factors and activation energies of the kinetic parameters of co-

polymerization of ethylene and a comonomer (common values for both systems) (𝒌𝒊 𝒌𝒊,𝟎𝒆𝑬𝒂𝑹𝑻)

Parameter Value

Spontaneous activation [37]

𝑘 , s-1 7.2104

𝐸 Jmol-1 33472

Spontaneous deactivation 15

𝑘 , s-1 7.2

𝐸 J mol-1 33472

Initiation 37

𝑘 , m3 mol-1 s-1 2.9102

𝐸 , J mol-1 37656

Spontaneous chain transfer 15

𝑘 , m3 mol-1 s-1 7.2

𝐸 J mol-1 33472

18

18

Transfer to hydrogen 37

𝑘 , m3 mol-1 s-1 6.3

𝐸 J mol-1 33472

Transfer to monomer [37]

𝑘 , m3 mol-1 s-1 0.15

𝑘 , m3 mol-1 s-1 0.43

𝑘 , m3 mol-1 s-1 0.15

𝑘 , m3 mol-1 s-1 0.43

𝐸 , J mol-1 33472

Activation energy of propagation [15]

𝐸 J mol-1 37656

Table 7. Propagation rate coefficients of the co-polymerization of ethylene with a comonomer

1-Butene 1-Hexene

Reactivity ratios (-):

𝑟 𝑘 /𝑘 42.5 18.94[38]

𝑟 𝑘 /𝑘 0.023 0.04 [38]

Pre-exponential factor of propagation m3 mol-1 s-1 :

𝑘 , 2.48104 [37] 2.48104

𝑘 , 5.82102 [37] 𝑘 /𝑟 1.3103

𝑘 , 1.86104[37] 𝑘 /𝑟 2.65103

𝑘 , 4.37102 1.06102 [38]*

Comonomer initiation m3 mol-1 s-1 :

𝑘 , 40.7[37] 𝑘 , 𝑘 /𝑘 1.25

𝐸 , J mol-1 37656 37656

* Calculated to respect the ratio 𝑘 /𝑘 in reference [38].

2.3 Fluidized bed reactor model

The FBR is modeled as a single-phase CSTR. This assumption is a reasonable initial

approximation in industrial FBR given its high recycle ratio and low single pass conversion.[8,

19

19

35] Fresh or prepolymerized catalyst particles, gas species (monomers, N2, H2) and condensed

gas (ICA) are assumed to be fed continuously at the bottom of the reactor. Thus, the FBR can

roughly be divided into two compartments: one super-dry compartment in the containing only

gas and polymer, and one much smaller compartment in the bottom also containing condensed

vapors. Only the upper compartment is considered in the present model, which represents most

of the reactor volume. Indeed, when operating under condensed mode, the injected liquid

species evaporate rapidly and the major part of the reactor only contains solid and gas species

(i.e. super-dry mode).[39]

The mass balances of the different species in the FBR are given in Table 8, with the following

notations, 𝜀 : porosity of the bed, 𝑋 and 𝑋 , : mass fractions of component i in the recycle

and the input stream respectively, ℎ(m): bed height, 𝑉 m : bed volume, 𝐴 m : cross-

sectional area of the bed 𝑄 (m3 s-1): bed outlet volumetric flow rate, 𝐹 (kg s-1) inlet flow rate

of component 𝑘, 𝐹 (kg s-1) recycling flow rate and 𝑋 mass fraction of component 𝑘 in the

gas phase.

Table 8: Mass balances of the different species in the FBR

Monomer i d Md𝑡

𝐹𝑀 𝜀 𝑉

𝑄 M𝑉

1 𝜀𝜀

𝑅M 𝐴𝑉

dℎd𝑡

Hydrogen d Hd𝑡

𝐹𝑀 , 𝜀 𝑉

𝑄 H𝑉

1 𝜀𝜀

𝑅H 𝐴𝑉

dℎd𝑡

Nitrogen d Nd𝑡

𝐹𝑀 , 𝜀 𝑉

𝑄 N𝑉

N 𝐴𝑉

dℎd𝑡

ICA d ICAd𝑡

𝐹𝑀 , 𝜀 𝑉

𝑄 ICA𝑉

ICA 𝐴𝑉

dℎd𝑡

Potential catalyst

sites S

d Sd𝑡

𝐹 S ,

𝜌 1 𝜀 𝑉𝑄 S

𝑉𝑅

S 𝐴𝑉

dℎd𝑡

Y: P , λ , λ , λ ,

𝜇 , 𝜇 , 𝜇

d Yd𝑡

𝑅𝑄 Y𝑉

Y 𝐴𝑉

dℎd𝑡

20

20

The mass balance for the bed height is given by:

dℎd𝑡

𝑅 𝑀 , 𝑅 𝑀 ,𝑉𝜌 𝐴

𝐹 /𝜌1 𝜀 𝐴

𝑄𝐴

(4)

And the steady state mass balance for the polymer in the bed is given by the following

equation:[40]

𝑄𝑅 𝑀 , 𝑅 𝑀 , 𝑉

𝜌𝐹 /𝜌1 𝜀

(5)

where 𝜌 and 𝜌 are the densities of the polymer (around 920 kg m-3) and catalyst (2800 kg

m-3), respectively. The dimensions of the bed are given in Table 9.

Table 9. Reactor dimensions

Parameter Designation Value

Dbed Bed diameter 4.75 m

𝜀 Bed voidage 0.7

h Height of the bed 13.3 m

2.4 Correlations of key properties

The main properties to be controlled in the gas-phase copolymerization of ethylene are the melt

index (𝑀𝐼, or melt flow index 𝑀𝐹𝐼, g/10 min) and the polymer density (𝜌 ). Correlations are

therefore needed to estimate these properties. The available correlations in the literature can be

divided into two categories:

A. Correlations which relate the final properties to the individual monomer conversions in the

reactor (i.e. to reacted species) (Table 10).[11] These correlations are universal and remain

valid when varying the operating conditions.

B. Correlations which relate the final properties to the operating conditions (i.e. T, P, or

concentrations of unreacted species in the gas phase), such as [11]:

21

21

ln 𝑀𝐼 3.5 ln 𝑘 𝑘 𝑘 𝑘 𝑘

𝜌 𝜌 𝜌 ln 𝑀𝐼 𝜌 𝜌

(6)

Where 𝑘 , 𝜌  are tuning parameters and M  and M  are comonomers. Other 

correlations also exist in the open literature.[14] [41–44] Such correlations are only valid for the

set of operating conditions for which they were derived and cannot be used to account for

thermodynamic effects such as the co-solubility effect. Therefore, such correlations are not

valid during grade transition where the operating conditions change.

In this work, the correlations of the first category will be employed (Table 10), as only such

correlations would be able to account for the co-solubility effect for instance. In this work, the

correlations developed by McAuley and MacGregor[11] for both MI and density are used (see

Table 10).

The derivation of these correlations is based on physical interpretations. For instance, the melt

index is highly correlated to the polymer molecular weight distribution and branching

characteristics.[45] To simplify, the instantaneous melt index is usually correlated to the

instantaneous average polymer molecular weight 𝑀 . The molecular weight is in turn affected

by the concentration of monomer, comonomer and hydrogen. More particularly, hydrogen

plays the role of a chain transfer agent in catalytic ethylene reactions, thus allowing to reduce

the polymer molecular weight.[46] It constitutes therefore a primary manipulated variable during

grade transitions. The melt index considered in this work represents the flow rate of a molten

polymer under the load of 2.16 kg at 190 °C (ASTM D1238).[47,26]

Concerning the polymer density, it is strongly affected by the length and the number of short

chain branches. Hence, it is mainly governed by the fraction of reacted comonomer in the co-

polymer (𝐶 ) (Table 10).[48] By creating short chain branches on the polymer, the comonomer

allows reducing the polymer density. As a consequence the crystallinity of the polymer also

22

22

decreases.[26] The comonomer constitutes the second important manipulated variable during

grade transition. The correlation proposed by McAuley and MacGregor[11] is based on patent

data collected by Sinclair [49], and it relates the instantaneous polymer density to the comonomer

incorporation in the polymer by including 𝐶 (where 𝐶 𝜑 100, and 𝜑 , see

Table 5).

Table 10. Correlations of Category A for the properties of the polymer: MI (the same relation is used for instantaneous and cumulative properties) and density

Ref Melt index (g/10 min) Density 𝐠 𝐜𝐦 𝟑 Data from

[11] 𝑀 kg mol 111525 𝑀𝐼

.

(or 𝑀𝐼 3.35 10 𝑀 . ) 𝜌 0.966 0.02386 𝐶 .

Butene grades [49]

[50] 𝑀𝐼 2.7 10 𝑀 . [51] 

[51,51] 𝑀𝐼 4.195 10 𝑀 .

[48] 𝑀𝐼 3 10 𝑀 . 𝜌 0.023 ln 𝐶 0.9192 Octene grades [53]

The cumulative properties (of the polymer exiting the reactor) can be calculated from the

instantaneous ones (those being produced at time t) by integration over the residence time (𝜏).

Therefore, the cumulative melt index (𝑀𝐼 ) becomes:[54]

d 𝑀𝐼 .

d𝑡1𝜏

𝑀𝐼 . 1𝜏

𝑀𝐼 . (7)

Similarly, the cumulative density of the polymer (𝜌 ) is given by: [8]

d 𝜌d𝑡

1𝜏

𝜌1𝜏

𝜌 (8)

3. Grade transition strategy

Dynamic optimization is employed to optimize the transition between different grades of

LLDPE in the FBR using the combined kinetic and thermodynamic model.

23

23

3.1 Formulation of the optimization problem

The manipulated variables are the flow rates of hydrogen and comonomer and the controlled

outputs are the melt index and the polymer density.

The optimization problem can be written as follows:[12]

min𝒖

𝐽 𝒖 𝑡 , 𝒙 𝑡 , 𝑡 ∈ 𝑡 , 𝑡 (9)

𝒖 𝒖 𝑡 𝒖 (10)

where J is the objective function, 𝒙 𝑡 is the vector of state variables (see Table 8) and 𝒖 𝑡

represents the vector of manipulated variables, 𝒖 𝑡 𝐹 , 𝐹 . The inequality constraints

(10) indicate the available ranges of manipulated variables.

3.2 Objective function

The considered objective function is the following:

𝐽 𝒖 𝑤 𝑤 𝑤 𝑤 d𝑡 𝑤 ∑ (11)

By considering both instantaneous and cumulative properties (of the melt index and polymer

density), and by a good tuning the weighting factors 𝑤 (with 𝑖 1 5), one may accelerate

the convergence of cumulative properties while keeping the instantaneous properties within an

acceptable range. The last term on the right hand side of this equation is also intended to

minimize the variation of the input during the transition, in order to avoid oscillations (as in

model predictive control). The normalization of the different terms (i.e. the division by the set-

points of the MI and density) allows for an easier tuning of the weighting factors. The indices

i, c and sp refer to the instantaneous, cumulative and set-point properties, respectively.

At a constant reaction rate, the proposed objective function allows reducing the quantity of

transition product as well as the transition time, even though the time is not explicitly minimized

24

24

in this function. If the reaction rate varies during the transition, then this objective function

allows minimizing only the transition time. The minimization of the transition product would

necessitate, in case of variable reaction rate, to multiply the criterion by the instantaneous

reaction rate (Rp), as done by McAuley and MacGregor[12] for instance. Takeda et al.[13]

suggested that the choice between minimizing the transition product and the transition time

should be based on the market demand: where at high polymer sales and plant capacity

production it is preferred to minimize the transition time; while at low polymer sales and

reduced plant capacity it is preferred to minimize the transition production and authorize a

longer transition time. A transition product can usually be sold, although at a discounted price.

3.2 Degrees of freedom of the inputs

It is usually sufficient to assume the manipulated variables (here, the flow rates of hydrogen

and comonomer) to vary by a series of ramps during the transition.[12] Based on the literature

study and the residence time of the FBR (4-6 hours in this study, depending on the operating

conditions), the transition is divided into 5 intervals, where the final interval corresponds to the

steady state interval, to maintain until the end of the production of the new grade. Thus, the

optimization allows switching the flow rates every 2 hours during the first 8 hours, and the last

ramp corresponds to the steady state flow rate of the new grade.

4. Simulation results and discussion

The proposed strategy is evaluated in grade transition starting from grade A, to grade B with

higher or lower MI and , then coming back to grade A, for both of the copolymerization

systems (Table 11). These choices are based on LLDPE specifications, i.e. MI[0.01-100]

g/10min [55] and [915-935] kg m-3 [56]. The duration of each grade production is usually

defined by the market demand, the specifications or the claims of the production. Here, an

arbitrary duration of production of each grade of 30 hours is implemented in both systems. No

25

25

particular change is required at the optimization level to change to shorter or longer production

periods, only the final times need to be indicated. The initial steady state conditions, producing

grade A, are given in

Table 12. This weighting factors were tuned as follows, except otherwise mentioned: w1 = 0.08,

w2 = 1, w3 = 8 w4 = 19 and w5 = 104. This choice was based on few simulation tests, in a way

to ensure a compromise between fast convergence of the cumulative properties while reducing

the overshoots of the instantaneous properties. Indeed, while allowing for big variations in the

instantaneous properties leads to a faster convergence of the cumulative properties, some

conditions of comonomer or hydrogen pressures might lead to polymer softening or sticking

problems.[3] In the following simulations, temperature control is assumed to be perfect in the

bed, so that the working temperature is constant.

Table 11. LLDPE grades considered in the grade transition policy

Grade Melt index Target (g / 10 min) Density Target (kg m-3)

1-hexene 1-butene 1-hexene 1-butene

A 0.54 4.5 923 918.5

B 2 2 916 922

A 0.54 4.5 923 918.5

Table 12. Initial conditions of the grade transition simulations (leading to grade A under steady state)

1-hexene 1-butene

T (°C) 90 90

𝑃 (bar) 9.4 7

𝑃 (bar) 0.35 1.55

PICA (bar) 0.6 3.5

𝑃 (bar) 2.2 2.2

26

26

4.1 Copolymerization of ethylene and 1-hexene in presence of n-hexane as ICA

The optimization strategy was evaluated using the parameters of the first system, i.e. the

copolymerization of ethylene with 1-hexene in presence of n-hexane. Note that the

thermodynamic model was developed for this system for ethylene pressure of 10 bar and

pseudo-component (i.e. comonomer plus ICA) pressure on the range of 0-1 bar. Therefore, the

simulations (including the choices of the set-points) are conducted in a way to respect these

ranges.

Figure 5 shows the results of the two grade transitions, from A to B, and from B back to A. This

scenario was simulated using the following weighting factors: w1 = 0.08, w2 = 1, w3 = 8 w4 =

19 and w5 =0, therefore the instantaneous properties (𝑀𝐼 and 𝜌 ) go beyond the set-point (SP)

during the transition in order to accelerate the convergence of the cumulative properties

(𝑀𝐼 and 𝜌 ). This is related to the variations of the flow rates of hydrogen and 1-hexene, which

are higher at the beginning of the grade transition and then they stabilize, as indicated by the

increase in the pressure. The overshoots in the instantaneous properties can be reduced by

reducing the weighting factors multiplying the cumulative properties w2 and w4 compared to

those of the instantaneous properties w1 and w3, or by considering w5 0, or by adding

constraints on the outputs, as discussed in the following scenarios. The MI is inversely

proportional to the polymer molecular weight. Therefore, an increase in the hydrogen pressure

during the transition, from grade A to B for instance, led to a decrease in the polymer molecular

weight and thus to an increase in the melt index. Likewise, an increase in the comonomer

pressure during the transition from grade A to B, led to an increase in the amount of short

branches and thus to a decrease in the polymer density. The proposed strategy allows to move

either to higher (grade A to grade B) or lower (B to A) values of 𝑀𝐼, and vice versa for 𝜌.

27

27

Figure 5.Grade transition in ethylene-1-hexene copolymerization in presence of n-hexane at 90°C (w1 = 0.08, w2=1; w3=8, w4 = 19, w5 = 0).

The figure shows that the concentration of ethylene in the polymer particles (which constitutes

the site of the reaction) does not change significantly during the transition, where the

comonomer flow rate is varied, so the co-solubility effect is negligible in this sense and under

the realized changes in the comonomer pressure. Note that the ethylene pressure is maintained

constant in all the grades. However, the concentration of comonomer in the polymer particles

is highly affected by these changes, which demonstrates the necessity of using a good

thermodynamic model. The impact of the thermodynamic model is investigated more deeply in

the last section. Note that the total pressure of comonomer and ICA reached 1.35 bar at the

maximum in this simulation, but only for a short duration, and therefore the employed

thermodynamic correlation remains valid during most of the time (PICA+Pcom=0-1 bar).

Mel

t ind

ex (

g / 1

0 m

in)

Pre

ssur

e (b

ar)

Po

lym

er M

w (

kg/m

ol)

(mol

m-3

pol

ymer

)

Pre

ssur

e (b

ar)

Pol

ymer

den

sity

(kg

m-3

)

28

28

Figure 6. Grade transition in ethylene-1-hexene co-polymerization in presence of n-hexane: Tracking of the cumulative properties (w1=w3=w5=0; w2=1; w4=4), with constraints on the

instantaneous properties (0.1<MIi<3 and 910<i<935).

The same scenario presented in Figure 5Figure 7. Grade transition in ethylene-1-butene co-

polymerization in presence of iso-butane (w1 = 0.08, w2=1, w3=8, w4 = 19 and w5 = 0). was

simulated while tracking only the cumulative properties (i.e. w1=w3=0) and considering

constraints on the instantaneous properties, as follows: 0.1<MIi<3 and 910<i<935 (Figure 6).

It can be seen that the convergence of the cumulative properties is slowed down compared to

Figure 5, but the overshoots in the instantaneous properties (𝑀𝐼 and 𝜌 ) are reduced and kept

within the constraints. Adding hard constraints allows remaining within the acceptable range

of properties, but it slows down the calculation. Another way to reduce the overshoots in the

instantaneous properties, without considering constraints, consists of increasing the values of

w2 and w4 with respect to w1 and w3 or by considering w5 0.

0 30 60 90Time (h)

0

1

2

3

4

Mel

t ind

ex (

g / 1

0 m

in) SP

MIc

MIi

0 30 60 90Time (h)

0

2

4

6

8

10

Pre

ssur

e (b

ar) Ethylene

H2

0 30 60 90Time (h)

0

50

100

150

200

Po

lym

er M

w (

kg/m

ol)

Mwc

Mwi

0 30 60 90Time (h)

0

100

200

300

400

500

(mol

m-3

pol

ymer

)

EthyleneComonomer

0 30 60 90Time (h)

0

0.2

0.4

0.6

0.8

1

Pre

ssur

e (b

ar)

PICA

Pcom

0 30 60 90Time (h)

900

920

940

960

Pol

ymer

den

sity

(kg

m-3

) SP

c

i

29

29

4.2 Copolymerization of ethylene and 1-butene in presence of iso-butane as ICA

The proposed grade transition optimization strategy was evaluated in the second system:

ethylene and 1-butene copolymerization in presence of iso-butane as ICA. Note that the

thermodynamic model was developed for different conditions for this system: i.e. ethylene

pressure of 7 bar and pseudo-component pressure on the range of 5-10 bars. The set-points of

the melt index and polymer density of grades A and B were also set differently in this system,

but still within LLDPE grades. The same weighting factors as the first system were considered.

Figure 7. Grade transition in ethylene-1-butene co-polymerization in presence of iso-butane (w1 = 0.08, w2=1, w3=8, w4 = 19 and w5 = 0).

Figure 7 shows the simulation results. The melt index converges in about 6 hours to the set-

point, while the density converges to the set-point in 2 hours. The overshoots of the

instantaneous properties remain acceptable, but they can be reduced by either manipulating the

weighting coefficients (as discussed in the following scenario) or by adding hard constraints on

0 30 60 90Time (h)

0

2

4

6

8SPMI

c

MIi

0 30 60 90Time (h)

0

2

4

6

8

10EthyleneH

2

0 30 60 90Time (h)

0

50

100

150

Mwc

Mwi

0 30 60 90Time (h)

0

100

200

300

400

500EthyleneComonomer

0 30 60 90Time (h)

0

1

2

3

4

PICA

Pcom

0 30 60 90Time (h)

910

915

920

925

930 SP

c

i

30

30

the outputs as discussed in the previous section. Note that the total pressure of comonomer and

ICA is around 5 bar, therefore the employed thermodynamic correlation is valid (PICA+Pcom=5-

10 bar).

The last term of the objective function (𝑤 ∑ ) can allow minimizing the variation of

the manipulated variables (flow rates of hydrogen and the comonomer), and thus to reduce the

overshoots in the instantaneous properties. Indeed, injecting big amounts of hydrogen or

comonomer rapidly increases the risk of polymer softening and stickiness.[3] As a consequence,

adding this term is expected to reduce the overshoots in instantaneous properties. Due to the

low values of the variations of the flow rates, it was necessary to have a high weighting factor,

w 10 , to ensure an impact on the performance. Figure 8 shows the results when adding

this term to the objective function, which is to be compared to Figure 7 done under the same

conditions but without this term. The figure clearly shows that the pressures of hydrogen and

comonomer undergo less changes. As a consequence, the instantaneous properties have lower

overshoots. However, this delays a little the convergence of the cumulative properties to the

set-points. A compromise is thus to be determined between fast convergence of the cumulative

properties and less variation in the instantaneous properties.

31

31

Figure 8. Grade transition in ethylene-1-butene: effect of the term 𝒘𝟓 ∑ 𝜟𝒖𝒊

𝒖𝒊 𝒕𝟎

𝟐𝒊 𝟏 in the objective

function (w1 = 0.08, w2=1, w3=8, w4 = 19 and w5 = 104)

In order to evaluate the gain realized by the optimization strategy, its performance was

compared to the case of injecting the optimal feed rates of the final grade during the transition

(here called the final steady-state, SS), as done for instance by Rahimpour et al.[8] (Figure 9).

When employing a constant flow rate during the transition, the convergence time is that of the

residence time of the reactor. It can be seen that employing the optimized varying flow rates

during the transition allows to reduce the convergence times of the cumulative melt index and

the density.

0 30 60 90Time (h)

0

2

4

6

8SPMI

c

MIi

0 30 60 90Time (h)

0

2

4

6

8

10EthyleneH

2

0 30 60 90Time (h)

0

50

100

150

Mwc

Mwi

0 30 60 90Time (h)

0

100

200

300

400

500EthyleneComonomer

0 30 60 90Time (h)

0

1

2

3

4

PICA

Pcom

0 30 60 90Time (h)

910

915

920

925

930 SP

c

i

32

32

Figure 9. Grade transition in ethylene-1-butene copolymerization: comparison between the proposed grade transition strategy (leading to varying optimized flow rates during the transition) and injecting a constant flow rate during the transition (corresponding to the optimal flow rate of

the final grade, under steady-state conditions)

4.3 Impact of the thermodynamic model

In order to demonstrate the importance of employing a good thermodynamic model in the

optimization strategy, two scenarios were simulated. The first scenario was performed by

assuming an error in the parameters of the thermodynamic model. The second system was used

for this purpose, i.e. ethylene-1-butene co-polymerization in presence of iso-butane as ICA.

In Figure 10, an error is assumed in the parameters A and E in equations 1 and 2, related to the

calculation of ethylene and comonomer concentrations in polymer, M and M ,

respectively. It can be seen that the employed flow rates bring the process to different set-points

than the desired ones (lower MI and , as the used parameters A and E were assumed to be

underestimated). Indeed, using lower A and E parameters in the model gives lower M and

0 30 60 90Time (h)

0

2

4

6

8

10MI

c

MIi

MIc-SS

MIi-SS

SP

0 30 60 90Time (h)

910

915

920

925

930c

i

c-SS

i-SS

SP

0 30 60 90Time (h)

0

0.5

1

1.5

210-5

OptimizedFinal steady-state

0 30 60 90Time (h)

0.6

0.8

1

1.2

1.4

1.610-3

OptimizedFinal steady-state

33

33

M than the real ones. In order to correct ratios of monomer to hydrogen as well as to

comonomer (to obtain the desired properties), the optimization strategy forces the decrease in

the hydrogen flow rate, which leads to an increase in the polymer molecular weight, and a

decrease in the MI. Similarly, the optimization forces the decrease in the comonomer flow rate,

and as a result of errors in M and M , a decrease in the polymer density is observed in this

case). Note that the optimization strategy continues to work adequately, but as it is based on a

wrong model it does not converge to the correct optimal points, therefore the use of an adequate

thermodynamic model is essential.

Figure 10. Influence of the thermodynamic model parameters on the process response, in ethylene and 1-butene copolymerization (Process parameters: A =1.98 mol m-3 bar-1, E=180 mol m-3, Model

parameters used for optimization: A =0.992 mol m-3 bar-1, E=90.2 mol m-3)

The second scenario was performed by switching to a binary model to describe the solubility

of the different species in the polymer (i.e. with no co-solubility effect). The system ethylene-

1-hexene co-polymerization in presence of n-hexane as ICA was used for this simulation. In

this case, the thermodynamic model leads to the calculation of an incorrect concentration of

ethylene in the amorphous phase of the polymer, M =257 mol m-3 at 10 bar ethyleneand 90°C

(as it assumes a binary system)[27] instead of around 280 mol m-3 estimated in the pseudo-

quaternary system. It also calculates an incorrect comonomer concentration in the polymer

0 30 60 90Time (h)

0

2

4

6

8

Mel

t in

dex

(g

/ 10

min

)

SPMI

c

MIi

0 30 60 90Time (h)

910

915

920

925

930P

oly

me

r d

ensi

ty (

kg m

-3)

SP

c

i

34

34

particle M . The concentration of 1-hexene in a binary system (1-hexene+LLDPE)[29] is

expected to be higher compared to its concentration in a ternary system due to the anti-solvent

effect of ethylene (as shown in Figure 2). However, combining ICA+comonomer in a pseudo-

quaternary system leads to a global pressure which is much higher. As indicated by Figure 4, a

small change in the pressure leads to a high change in the solubility of the pseudo-component,

so the quaternary system leads to a much higher concentration of comonomer in the particles

than in a binary system at the same pressure. Note however that the same flow rate is injected

in the model and the process, but different reaction rates occur (due to the use of different

thermodynamic models), therefore the comonomer pressure varies a lot between the two

simulations, and therefore it is not straightforward to compare the concentration of comonomer

in the model and the process in this simulation. In this simulation, when M = 165 mol m-3 in

the pseudo-quaternary system, it was M = 159 mol m-3 in the binary model.

The simulation test was performed using the binary model for both the concentrations of

ethylene and 1-hexene in the amorphous phase of the polymer (so the model and grade

transition is simulated using the binary model while the process is simulated using the pseudo-

quaternary model). Figure 11 shows that using a binary thermodynamic model and ignoring the

co-solubility effect leads to a big drift of the properties from the set-points. Indeed, the model

assumes a lower M (so a lower polymer molecular weight and a higher MI). Therefore, the

optimization strategy based on this model makes the decision to decrease the hydrogen flow

rate. However, when implemented to the process (simulated using the pseudo-quaternary

model, where the concentration of monomer is higher), this flow rate leads to a higher MW, so

to a lower MI. Following the same reasoning, a drift in the polymer density occurs due to errors

in both M and M . This simulation demonstrates the importance of using an adequate

thermodynamic model in the optimization strategy.

35

35

Figure 11. Influence of using binary thermodynamic model (not taking in account the co-solubility effect) on the process response. System ethylene-1-hexene copolymerization in presence of n-

hexane at 90°C. (w1 = 0.08, w2=1, w3=8, w4 = 19 and w5 = 104)

5. Conclusions

In this work, off-line dynamic optimization was implemented to optimize the grade transitions

in a fluidized bed reactor of polyethylene. A combined kinetic and thermodynamic model was

used in order to account for the co-solubility effects of the different gas species. The

thermodynamic model is based on Sanchez-Lacombe EoS, but then simplified correlations

were used to reduce the calculation time. Two copolymerizations were considered, the

copolymerization of ethylene with 1-hexene in presence of n-hexane as ICA and the

copolymerization of ethylene with 1-butene in presence of iso-butane. Some assumptions were

made, mainly due to the lack of thermodynamic data in the literature, to allow the prediction of

the solubility of the different species in PE in these quaternary systems.

The simulation results demonstrate the importance of the thermodynamic model in the

optimization strategy. A good control of the polymer melt index and density could be realized

by manipulating the flow rates of hydrogen and comonomer. Nevertheless, in both systems, the

co-solublity effect of comonomer on ethylene was not observed, which is due to the low impact

of the pseudo-component on the solubility of ethylene under the employed operating conditions

0 30 60 90Time (h)

0

1

2

3

4M

elt

ind

ex (

g /

10 m

in)

SPMI

c

MIi

0 30 60 90Time (h)

910

920

930

940

950

960

Po

lym

er

den

sity

(kg

m-3

)

SP

c

i

36

36

(pressure and temperature). The importance of the thermodynamic model was mainly related

to evaluating the concentration of comonomer in the polymer during the transition, which

highly impacts the polymer properties.

Both the instantaneous and the cumulative properties could be controlled, in a duration much

lower than the residence time of the reactor. The role of the weighting factors, in the

minimization function, is determinant at this level, where it can either give more importance to

controlling the instantaneous properties (thus eliminating any overshoot) or on the contrary

allow a faster control of the cumulative properties in detriment of the instantaneous ones. A

compromise between these two options is necessary in order to ensure a fast convergence of

the cumulative properties to the set-points (thus reduce the transition product) while avoiding

big variations in the flow rates or pressures of hydrogen and comonomer as they may increase

the risk of polymer sticking or softening. To reach the same objective, constraints on the

instantaneous properties can be considered, but this slows down the calculation.

The proposed optimization tool should allow a more efficient operation and a better control of

the polymer quality. The kinetic parameters and the correlations of the polymer properties used

in this work were taken from literature, as well as the assumption of the bed to behave as a

CSTR. These models can be replaced by more detailed models when available in the same

optimization strategy. For instance, improvement at the level of the bed model can be done by

considering a compartmental model and at the kinetic level by considering multiple site

catalysts leading to bimodal molecular weight distributions and using correlations of the MI

and polymer density adapted to such systems. Finally, the availability of more thermodynamic

data or the use of a particle model accounting for diffusion would allow to improve the precision

of the optimization strategy outcome.

37

37

Acknowledgements

This work was funded by Agence Nationale de la Recherche (Thermopoly project, grant N°

ANR-16-CE93-0001-01). The authors would like to thank M. Robert Pelletier for his input on

grade transitions in polymerization processes.

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